Weakly invariant sets of hybrid systems

We consider a mathematical model of a hybrid system in which the continuous dynamics generated at any point in time by one of a given finite family of continuous systems alternates with discrete operations commanding either an instantaneous switching from one system to another, or an instantaneous passage from current coordinates to some other coordinates, or both operations simultaneously. As a special case, we consider a model of a linear switching system. For a hybrid system, we introduce the notion of a weakly invariant set and analyze its structure. We obtain a representation of a weakly invariant set as a union of sets of simpler structure. For the latter sets, we introduce special value functions, for which we obtain expressions by methods of convex analysis. For the same functions, we derive equations of the Hamilton-Jacobi-Bellman type, which permit one to pass from the problem of constructing weakly invariant sets to the control synthesis problem for a hybrid system.